Cavity enhancement methods, systems and devices, and methods of measuring same

ABSTRACT

A system for increasing light throughput in cavity enhanced spectrometry, and a model for cavity enhanced absorption measurements are presented. The cavity has an entrance mirror, an opposed exit mirror and a detector positioned adjacent the exit mirror. An input aperture is defined in the entrance minor to allow light from a source to enter the cavity. The input aperture improves light throughput without significant departure from the theoretically predicted amplification of absorbance. This results in improvement of detection limits, even with mirrors of modest reflectivity and inexpensive detectors.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The invention that is the subject of this application was made with U.S.government support under NSF EAGER grant CHE-1246368 awarded by theNational Science Foundation. The U.S. government may have certain rightsin this invention.

FIELD OF THE INVENTION

This invention relates generally to methods, devices and systems forincreasing light throughput in cavity enhanced spectrometry, and a modelfor cavity enhanced absorption measurements. More specifically, cavitymethods, devices and systems are provided to increase light throughputin the cavity without significant loss of cavity enhancement resultingin improvements of detection limits.

BACKGROUND OF THE INVENTION

Absorption spectrometry is arguably the most commonly used quantitationtechnique used for analysis. To increase detection sensitivity, onechoses the optimum chemistry and the measurement wavelength(absorptivity c at maximum). Absorbance also increases with thepathlength. In the gas phase, single long paths as in differentialoptical absorption spectroscopy or one attained through multi-reflectioncells can be used. Simply increasing the absolute value of theabsorbance is not tantamount to improving the limit of detection (LOD),however. In the liquid phase, due to increased beam divergence, light israpidly lost to the wall and the relative noise increases concomitantlyas the detector becomes light starved. Thin walled straight glass tubesor those filled with high refractive index (RI) organic solvents helpguide light via total internal reflections at the glass-air andsolvent-glass interfaces, respectively; neither system is particularlyuseful: in one case deposition of any particles on the glass outersurface causes loss of light, in the other case, water is the solvent ofinterest in majority of applications. A gas shell can be formed around aliquid in a hydrophobic porous membrane tube but a gas bubble is tooreadily formed. Aqueous solutions containing a lot of solute or ethanolcan have an RI greater than that of FEP Teflon® 1.34 and thus a FEP tubefilled with such a solution also behaves as a liquid core waveguide(LCW). LCWs with a purely aqueous solution as the core have becomepractical only after the introduction of Teflon AF (RI 1.29-1.31,compared to 1.33 for water).

In all the above arrangements, the incident light assumedly traces asingle path; these are both single path and single-pass cells, i.e., thelight does not trace the same path twice. Such cells have a singleeffective overall pathlength for the purposes of Lambert-Beer's lawregardless of the cell contents. If a particular chemistry—wavelengthcombination leads to a certain dynamic range, changing the pathlengthcan merely change the location of the usable range in the concentrationdomain.

In contrast, a multipath arrangement behaves differently. If theshortest path that the light can travel to reach the detector is b, theeffective pathlength (henceforth designated b) is absorbance dependent.It can be perceived that at high absorbances, longer paths contributeslittle to the transmitted light and

${\lim\limits_{A = \infty}\overset{\_}{b}} = {b.}$

the value of b at the lower absorbance limit is of greater interest asit is often the determinant of the concentration limit of detection(LOD). For a multiplicity of pathlengths b_(i), each having f_(i)fraction of the total light (Σf_(i)=1), it has been shown that b at thelower absorbance limit is simply the weighted sum b=Σf_(i)b_(i). For anevenly illuminated wedge shaped cell, for example, at the lowerabsorbance limit b can be half the base width of the cell.

The combination of cells with different pathlengths, as in awedge-shaped cell, is not practical; moreover,

$\lim\limits_{A = \infty}\overset{\_}{b}$

can still be less than the longest physical pathlength. Puttingpartially reflective mirrors on both the entrance and exit windows ofany conventional cell can serve the same purpose. Such a system,hereinafter called a single-path multipass cell (multiple reflections onthe same path as the beam traverses back and forth, each time losinglight both due to absorption by the medium and transmission through thepartially transmissive mirrors), is provided in order to increase thepathlength, especially at the low absorbance end.

It can be shown that the gain in pathlength at the low absorbance limitis equal to

$\frac{1 + R^{2}}{1 - R^{2}},$

where R is the reflectivity of the mirror (R being 1 for a perfectlyreflective minor and 0 for a perfectly transparent object). It can befurther shown that for values of R approaching 1

$,\frac{1 + R^{2}}{1 - R^{2}}$

is well approximated by

$\frac{1}{1 - R}.$

Thus for example, a cell with a physical pathlength of 1 cm, bounded bymirrors of 99% reflectivity (R=0.99) will have an effective path lengthof

$\overset{\_}{b} = {\frac{1}{1 - r} = {1000\mspace{14mu} {{cm}.}}}$

While an increase in the mirror reflectivity R increases the pathlengthand a proportionate amplification of absorbance, increases in R alsoresults in lower light throughput and increases the relative noise ofthe detector. With R=0.99 minors for example, with 10,000 photonsincident on the entrance minor, only 100 photons will make into the celland if there is no attenuation by the solution in the cell, of 100photons making it to the exit mirror only 1 will exit to reach thedetector. What is needed then is a means for increasing light throughputwithout significant loss of cavity enhancement and reducing the need forincreased source brightness for sufficient light to reach the detector.

SUMMARY

Even with coherent light sources, there can be finite beam divergence.Presented herein are methods, devices and systems for increasing lightthroughput in cavity enhanced spectrometry in the presence of beamdivergence, and a model for cavity enhanced absorption measurements.

In one aspect, a measurement cell comprises a partially reflectiveentrance mirror and a partially reflective exit mirror defining a cavitytherebetween. In another aspect, the entrance minor can be placed onand/or form at least a portion of a first side of the cell, and the exitmirror can be placed on and/or form at least a portion of a second sideof the cell.

An input aperture can be defined in at least a portion of the entranceminor. In one aspect, while the aperture can be small in an absolutescale, the aperture can be sized and shaped such that the source lightcan be efficiently and effectively entered into the cell through theaperture. At least a portion of the light entering the cell through theaperture can travel across the cavity and reflect off the exit mirrorand back towards the entrance mirror. A portion of this reflected lightcan be lost from the cell through the aperture defined in the entranceminor, the extent of the loss depending on the beam divergence. For adivergent beam, when the aperture is made small and the light is focusedsuch that the beam waist enters the aperture (or the divergent source issituated at the aperture), this loss can be small compared to the totalbeam area. By introducing light into the cell through the aperturedefined in the entrance minor, the overall light throughput in the cellcan be increased. As an approximate guide line, with R=0.99 minors, ifthe light enters through the aperture rather than through a mirror, 100times as much light can enter the cell. The relative noise decreaseswith the square root of the light intensity; as such, a 10-fold decreasein the noise can be observed.

In use, light emitted from a light source can enter the cell through theinput aperture and can be detected by a detector configured to convertlight detected into an electrical signal for further analysis by acomputer or other processor. The input aperture, in one aspect, canimprove light throughput in the cell without significant departure fromthe theoretically predicted amplification of absorbance. Further, theinput aperture can improve light throughput in the cell such thatdetection limits of the detector are improved, even when used withminors of modest reflectivity and inexpensive detectors.

Related methods of operation are also provided. Other apparatuses,methods, systems, features, and advantages of the cavity enhancementsystem and the method of its use will be or become apparent to one withskill in the art upon examination of the following figures and detaileddescription. It is intended that all such additional apparatuses,methods, systems, features, and advantages be included within thisdescription, be within the scope of the cavity enhancement system andthe method of its use, and be protected by the accompanying claims.

DESCRIPTION OF THE FIGURES

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate certain aspects of the instantinvention and together with the description, serve to explain, withoutlimitation, the principles of the invention Like reference charactersused therein indicate like parts throughout the several drawings.

FIG. 1 is a view of a single path multipass cell arrangement,schematically shown, according to one aspect. The individual ray tracesare all actually perpendicular to the minors and are superimposed oneach other. They are shown in this manner to indicate the intensityafter each event. Similarly, a single beam emerges to fall on thedetector. Minor reflectance is R, transmittance is T, extinctioncoefficient of the medium is α.

FIG. 2 illustrates graphically the difference between the O'Keefe(equation 5) and the geometric series summation (equation 9) approachesin predicting b as a function of α for various values of R.

FIG. 3A is a graph illustrating the effective pathlengths computed forthe cell arrangement in FIG. 3B. In this off axis-arrangement, bothentrance and exit apertures are present to increase light throughput.The entrance and exit sides (in this aspect, the enclosure) act asreflective minors. L is the vertical distance between the entrance andexit point, Φ is the angle between the vertical distance ‘L’ and theshortest path length (considered as unity for standard cells), θ is theangle between the vertical distance ‘p’ and the shortest path length and‘h’ is the distance travelled by the incident light at an angle θ. Notelogarithmic axes in the plot. Note also that the shortest sourcedetector distance is √{square root over (2)} if this is considered thebase path length; the path length amplification factor is ˜40% less thanthe ordinate values shown.

FIG. 3B is schematic view of a single path multipass cell arrangement iswhich a source is offset from a detector, according to one aspect.

FIG. 4 graphically illustrates the effect of finite number of reflectionterms summed in equation 7 for different values of the extinctioncoefficient α on the effective pathlength b. If FIG. 4 b is plotted witha logarithmic abscissa, qualitatively it can have the same shape asFIGS. 2 and 3.

FIGS. 5A and 5B are graphs illustrating Beer's law behavior ofBromocresol Green at pH 12 in conventional linear as well as logarithmicaxes.

FIGS. 6A and 6B are graphs illustrating the experimental values of b asa function of Bromocresol Green concentrations with different cell typesand (a) CCD spectrometer detector, (b) UV 3600 Spectrophotometerdetector, R=0.99 mirrors. The inset shows the same data but with theordinate on a logarithmic scale.

FIG. 7 is a schematic view of a cell arrangement comprising a liquidcore waveguide and an optical fiber bundle, according to one aspect.

FIG. 8 is a schematic view of a cell arrangement comprising a liquidcore waveguide and an optical fiber, according to one aspect.

DESCRIPTION OF THE INVENTION

The present invention can be understood more readily by reference to thefollowing detailed description, examples, and claims, and their previousand following description. Before the present system, devices, and/ormethods are disclosed and described, it is to be understood that thisinvention is not limited to the specific systems, devices, and/ormethods disclosed unless otherwise specified, as such can, of course,vary. It is also to be understood that the terminology used herein isfor the purpose of describing particular aspects only and is notintended to be limiting.

The following description of the invention is provided as an enablingteaching of the invention. Those skilled in the relevant art willrecognize that many changes can be made to the aspects described, whilestill obtaining the beneficial results of the present invention. It willalso be apparent that some of the desired benefits of the presentinvention can be obtained by selecting some of the features of thepresent invention without utilizing other features. Accordingly, thosewho work in the art will recognize that many modifications andadaptations to the present invention are possible and can even bedesirable in certain circumstances and are a part of the presentinvention. Thus, the following description is provided as illustrativeof the principles of the present invention and not in limitationthereof.

As used herein, the singular forms “a,” “an” and “the” include pluralreferents unless the context clearly dictates otherwise. Thus, forexample, reference to an “emitter” includes aspects having two or moreemitters unless the context clearly indicates otherwise.

Ranges can be expressed herein as from “about” one particular value,and/or to “about” another particular value. When such a range isexpressed, another aspect includes from the one particular value and/orto the other particular value. Similarly, when values are expressed asapproximations, by use of the antecedent “about,” it will be understoodthat the particular value forms another aspect. It will be furtherunderstood that the endpoints of each of the ranges are significant bothin relation to the other endpoint, and independently of the otherendpoint.

As used herein, the terms “optional” or “optionally” mean that thesubsequently described event or circumstance may or may not occur, andthat the description includes instances where said event or circumstanceoccurs and instances where it does not.

Terms used herein, such as “exemplary” or “exemplified,” are not meantto show preference, but rather to explain that the aspect discussedthereafter is merely one example of the aspect presented.

Presented herein, in one aspect, is a generally applicable mathematicalfoundation for cavity enhanced absorption measurements regardless of theexact values of R (the minor reflectance), T (the mirror transmittance)and a (the extinction coefficient of the medium). In another aspect, aninput aperture for the cavity has been provided which can be of value insituations wherein beam divergence is relatively significant. In thisaspect, the input aperture can result in greater light throughputwithout significant loss of cavity enhancement and thus can reduce theneed for increased source brightness for sufficient light to reach adetector.

Optics and the analytical chemistry literature respectively use naturaland base-10 logarithmic relationships to describe light extinction. Asused herein, the respective extinction coefficients will be denoted asα′ and α(α′=α ln(10)), the base-10 version of Lambert's law beingI=I₀10^(−αb), where I₀ and I are the total incident and transmittedlight intensities, and α is equal to ∈c, ∈ being the molar absorptivityof the solute and c its concentration. It is assumed that the base pathlength b on each side of which a partial minor can be placed to beunity. Since units are not specified, any conclusion can be equallyvalid for any other value of b. With b=1, b is not only the effectivepathlength, but it also can be the amplification factor of the physicalpathlength, sometimes referred to as the cavity enhancement factor. Itis also assumed that R+T=1, i.e., minor absorptance is negligible (note,however, that it can readily be shown that a finite absorptance has noeffect on the results, except for a reduction in the overall lightthroughput).

Although others have considered the theoretical aspects of single pathmultipass systems, the focus has been on the accurate determination ofa, the extinction coefficient of the medium in the cavity. An accuratemathematical description that adequately takes into account finiteabsorption by the intervening medium has not been given and is providedbelow.

FIG. 1 depicts a measurement cell 10, according to one aspect. Themeasurement cell comprises an entrance minor 12 and an exit mirror 14.In another aspect, at least one of the entrance mirror and the exitmirror can be a partially reflective minor. In a further aspect, theentrance mirror can be placed on and/or form at least a portion of afirst side 16 of the cell 10, and the exit minor can be placed on and/orform at least a portion of a second side 18 of the cell 10 such that acavity is defined between the first and second sides. At least a portionof the entrance mirror and/or the exit mirror can be a nonfocusingmirror. In another aspect, at least a portion of the entrance mirrorand/or the exit mirror can be substantially planar. The minor can beformed from a conventional mirror, a silvering solution, aluminizedTeflon, aluminized Mylar and the like. In use, light emitted from alight source 20 can enter the cell and can be detected by a detector 22configured to convert light detected into an electrical signal forfurther analysis by a computer or other processor.

Following the notations used above, with I₀ incident on the entrancemirror, I₀T enters the measurement cell 10. Light reaching the exitmirror 14 is I₀T10^(−α) of which I₀T²10^(−α) is transmitted andI₀TR10^(−α) is reflected. Light is again attenuated as it traverses thecell and reaches the entrance minor 12 with the intensity I₀TR10^(−2α).I₀TR²10^(−2α) is reflected back, I₀TR²10^(−3α) reaches the exit minor 14and I₀T²R²10^(−3α) is transmitted. The total transmitted light I can berepresented by the Infinite geometric series:

I=I ₀ T ²10^(−α)Σ(1+r+r ² +r ³ +r ⁴+ . . . )  (1)

where r=R²10^(−2α).

Recognizing that the sum of the geometric series in parentheses is(1−r)⁻¹, equation 1 reduces to:

$\begin{matrix}{I = \frac{I_{0}T^{2}10^{- \alpha}}{\left( {1 - {R^{2}10^{{- 2}\alpha}}} \right)}} & (2)\end{matrix}$

This leads to the expression of b:

$\begin{matrix}{\overset{\_}{b} = {\frac{1}{\alpha}{\log \left( \frac{1 - {R^{2}10^{{- 2}\alpha}}}{10^{- \alpha}\left( {1 - R^{2}} \right)} \right)}}} & (3)\end{matrix}$

Note that in deriving equation 3, there was no implicit assumption thatminor absorptance is negligible or that the reflectivity is necessarilyvery high. Equation 3 can be valid for values of R or values of R+T thatare less than and/or significantly less than 1. At the low absorbancelimit of equation 3, it can be readily derived that:

$\begin{matrix}{{\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}} = \frac{1 + R^{2}}{1 - R^{2}}} & (4)\end{matrix}$

The same system has previously been considered by Dasgupta and Rhee andO'Keefe. However, Dasgupta and Rhee used an interferometric model thatis inapplicable here because the bandwidth of the source and the qualityof the cavity will not be expected to hold in the present experiments.O'Keefe assumed negligible mirror absorptance and derived (a missingminus sign has been corrected)

$\begin{matrix}{I = {- \frac{I_{0}T^{2}^{- {\alpha\prime}}}{2\mspace{14mu} {\ln \left( {Re}^{- {\alpha\prime}} \right)}}}} & (5)\end{matrix}$

Equation 5 is normally used to describe cavity absorption behavior athigh values of R and low values of α. It is an approximation while thesummation of the transmitted light terms (these represent a geometricseries) depicted in FIG. 1 provides a globally correct solution inpredicting b for any value of a or R. O'Keefe did not explicitlycalculate the gain in path length but this can be computed from equation5 to be:

$\begin{matrix}{\overset{\_}{b} = {\frac{1}{\alpha^{\prime}}{\ln \left( \frac{{\ln (R)} - \alpha^{\prime}}{^{- {\alpha\prime}}{\ln (R)}} \right)}}} & (6)\end{matrix}$

At the low absorbance limit,

$\begin{matrix}{{\lim\limits_{\alpha^{\prime}\rightarrow 0}\overset{\_}{b}} = {1 - \frac{1}{\ln (R)}}} & (7)\end{matrix}$

A comparison of equations 3 and 6 for b as a function of a for variousvalues of R appear in FIG. 2. The figure is in logarithmic scale tocover a large span; unfortunately, such scaling also minimizes thevisual difference. Nevertheless, it is apparent that the expectationsfrom both equations are nearly the same at high values of R and lowvalues of α and there is significant difference when either R is low orα is high. It can also be noted that for either equation 3 or 6, at highvalues of R, the limiting value of b is approximately given by:

$\begin{matrix}{{\lim\limits_{{R\rightarrow 1},{\alpha^{\prime}\rightarrow 0}}\overset{\_}{b}} \cong \frac{1}{1 - R}} & (8)\end{matrix}$

In a nominally single-path multipass system of perpendicular beamincidence, there can be finite beam divergence and a finite detectorarea. As can be appreciated, all of the light coming out on the detectorside cannot be captured by the detector. Thus a real situation likelyrepresents a finite number of reflections. This can be simulated bytruncating the infinite series expression in equation 1 to any desirednumber of terms. The results for 1-100 terms summed for a standard 1 cmpath cell 10, for absorbance values ranging from 10⁻⁶ to 0.2 is shown inFIGS. 4 a and 4 b, according to one aspect. In this aspect, the entrancemirror 12 and the exit mirror 14 have a reflectance R=0.99. Note that“number of terms” is not synonymous with number of passes counted. Forexample, when three terms are counted, the first term results from asingle base path traverse, while the second and the third terms resultfrom three and five base-path traverses, respectively.

As may be intuitive (if the series is limited to a single term, that isstandard Beer's law behavior; b can be >1 only in the presence ofsubsequent terms), b increases with increasing number of terms summed(FIG. 4 a), although even after 100 terms b does not reach the limitpredicted by equation 4 (compare the FIG. 4 a vs. the FIG. 2 R=0.99trace). Given the same number of terms, b can also increase withdecreasing α (FIG. 4 b), much the same as the infinite series sum inFIG. 2, except that the sum obtained here is less than that fromequation 3. In the above considerations, it is assumed that the numberof terms that count and the extinction coefficient α are independentvariables. In reality, when light loss in each pass is significant(e.g., because of beam divergence and/or R is low, the effective numberof terms can decrease with increasing α). A typical detector does notcover all the area through which light can exit through the cell.Although in practice light is reflected around infinitely, effectivelyonly a finite number of reflections, the exact number depends on beamdivergence, R, α and the fraction of the total light exit area coveredby the detector.

In one aspect, light throughput in the measurement cell 10 can be animportant consideration. In a preferred situation, the detector can onlybe limited by shot-noise. For example, if there are temporal sourcefluctuations, it is assumed that this can be compensated for byreferencing. However, if light reaching the detector 22 is notsufficient (whether because R is too high and/or I₀ is too low),increasing the absorbance signal will not necessarily result in anyimprovement in lowering the limits of detection (LOD) which is typicallydefined on the basis of Signal/Noise (S/N)=3. Further,counterintuitively, putting minors of very high R on each side of a cellin a standard spectrophotometer can deteriorate the LOD, not improve it.

The signal S in the present case is defined as the difference in theamount of light reaching the detector 22 in the presence of an absorbingsample (α=α) from that when α=0. Adapting equation 2:

$\begin{matrix}{S = {{I_{\alpha = 0} - I_{\alpha = \alpha}} = {\frac{I_{0}T^{2}}{\left( {1 - R^{2}} \right)} - \frac{I_{0}T^{2}10^{- \alpha}}{\left( {1 - {R^{2}10^{{- 2}\; \alpha}}} \right)}}}} & (9)\end{matrix}$

Assuming zero absorptance (R+T=1) equation 9 can reduce to

$\begin{matrix}{S = \frac{{I_{0}\left( {1 - R} \right)}{\alpha^{\prime}\left( {1 + R^{2}} \right)}}{\left( {1 + R} \right)\left( {1 - {R^{2}\left( {1 - {2\alpha^{\prime}}} \right)}} \right)}} & (10)\end{matrix}$

Only for R approaching unity, at low values of α, equation 10 canfurther reduce to

$\begin{matrix}{S = \frac{I_{0}\alpha^{\prime}}{2}} & (11)\end{matrix}$

In a shot noise limited situation, the noise associated with each of theterms in equation 9 is proportional to the square root of each of theterms. As the noise can add in a root mean square fashion the expressionof the overall noise N can be written as:

$\begin{matrix}{N = {k\sqrt[{- \alpha}]{\frac{{I_{0}\left( {1 - R} \right)}^{2}}{\left( {1 - R^{2}} \right)} - \frac{{I_{0}\left( {1 - R} \right)}^{2}10}{\left( {1 - {R^{2}10^{{- 2}\alpha}}} \right)}}}} & (12)\end{matrix}$

where k is a constant of proportionality. This equation simplifies to:

$\begin{matrix}{N = {k\sqrt{\frac{1 + {\left( {1 - \alpha^{\prime}} \right)\left( {1 - {2\; R^{2}}} \right)}}{\left( {1 + R} \right)\left( {1 - {R^{2}\left( {1 - {2\alpha^{\prime}}} \right)}} \right)}}}} & (13)\end{matrix}$

For R approaching unity, at low values of α, equation 13 can furtherreduce to:

N=k√{square root over (I ₀(1−R))}  (14)

leading to the following expressions for S/N and LOD:

$\begin{matrix}{{S/N} = {0.5\frac{\alpha^{\prime}\sqrt{I_{0}}}{k\left( {1 - R} \right)}}} & (15) \\{{LOD} = {1.5\frac{\alpha^{\prime}\sqrt{I_{0}}}{k\left( {1 - R} \right)}}} & (16)\end{matrix}$

In one aspect, the above considerations apply for an incident light beamrepeatedly traversing the same path. Effects of beam etendue/divergenceor the dependence of the exact reflectance of high-R dielectric mirrorson the incidence angle are not accounted for. However, even for acoherent/collimated beam, some beam divergence is likely upon a largenumber of reflections. Previously, there have been no theoreticalconsiderations of multiple paths originating from reflections within anenclosure.

When a divergent light source 20 is used, in one aspect and withreference to FIG. 1, the light can be introduced into the cell 10through an aperture 24 defined in a portion of the entrance mirror 12.At least a portion of the light entering the cell through the aperturecan reflect off the exit minor 14 and back towards the entrance mirror.As can be appreciated, a portion of the reflected light can be lost fromthe cell through the aperture 24 defined in the entrance mirror. Thisloss, however, can be small compared to the total beam area of adivergent beam. In another aspect, by introducing light into the cellthrough the aperture defined in the entrance mirror, the overall lightthroughput in the cell can be increased by the factor(1−R_(entrance))⁻¹.

In one aspect, the aperture 24 defined in a portion of the entrancemirror can be substantially circular in cross-sectional shape. In otheraspects, the aperture can be substantially oval, substantiallyelliptical, or any other shape. The aperture 24 can have a substantiallyconstant diameter, according to one aspect. Alternatively, the diameterof the aperture can increase or decrease as the aperture extends from anouter surface 26 of the entrance minor to an inner surface 28 of theentrance mirror. In another aspect, the aperture 24 can have a diameteror other dimension (such as height and/or width) of less than about 0.1mm, about 0.1 mm, about 0.2 mm, about 0.3 mm, about 0.4 mm, about 0.5mm, about 0.6 mm, about 0.7 mm, about 0.8 mm, about 0.9 mm, about 1.0mm, about 1.1 mm, about 1.2 mm, about 1.3 mm, about 1.4 mm, about 1.5mm, about 1.6 mm, about 1.7 mm, about 1.8 mm, about 1.9 mm, about 2 mm,about 2.25 mm, about 2.5 mm, about 2.75 mm, about 3 mm, about 3.25 mm,about 3.5 mm, about 3.75 mm, about 4 mm, about 4.25 mm, about 4.5 mm,about 4.75 mm, about 5 mm, about 5.5 mm, about 6 mm, about 6.5 mm, about7 mm, about 7.5 mm, about 8 mm, about 8.5 mm, about 9 mm, about 9.5 mm,about 10 mm or greater than about 10 mm.

In one aspect, the cell 10 can comprise a cuvette 30, such as anexternally silvered cuvette as depicted in FIG. 3B and the like. In thisaspect, each wall of the cuvette can behave as a mirror. A divergentlight source 20, such as, for example and without limitation, a lightemitting diode (LED) can be used to input light into the cell throughthe aperture 24 defined in a portion of the entrance minor 12 of thefirst side 16 of the cell. In another aspect, the detector 22 can bepositioned on the opposed second side 18 of the cell located at adifferent horizontal plane relative to the source. That is, the locationof the detector can be positioned higher or lower relative to theaperture 24 defined in the entrance minor. In a further aspect, alongitudinal axis L_(D) of the detector can be offset a predeterminedamount from a longitudinal axis Ls of the source and/or a longitudinalaxis L_(I) of the input aperture.

The detector 22 can be positioned behind the exit mirror 14 to detectlight transmitted through the exit minor. Optionally, an exit bore 32can be defined in a portion of the exit mirror and/or the second side 18of the cell and the detector can be positioned behind or otherwiseadjacent the exit bore. In one aspect, as long as the detector size issmall relative to the beam dimensions from the source, the theoreticalconsiderations can differ little other than an increase in the lightthroughput by the factor (1−R_(exit))⁻¹.

Inside the measurement cell 10, in this aspect, a direct ray of lighttraveling from the source 20 to the detector 22 can be at an angle φrelative to the longitudinal axis L_(D) of the detector and thelongitudinal axis Ls of the source and/or the longitudinal axis L_(I) ofthe input aperture. For convenience, it is assumed that φ is π/4 rad,such that a vertical distance between the source and the detector is thesame as the horizontal base path b from the first side 16 of the cell tothe opposed second side 18 (b is again assumed to be unity). In anotheraspect, light can also travel by reflection being incident at some otherangle θ(0<θ<φ) and in doing so, the ray of light traverses a distance h,covering a vertical distance of p (equal to tan θ), as indicated. Toreach the detector 22 with a vertical rise of unity, these steps shouldoccur n times where n=1/p. The n steps result in a total beam traverseof 2hn; in the process, 2n−1 reflections take place. Note also that hnis equal to 1/sin θ. If I_(0,θ) is the initial intensity of the beamtraveling with the angle θ, the transmitted light intensity I_(θ) can begiven by:

I _(θ) =I _(0,θ) R ^(2n-1)10^(−2hnα)  (17)

The total transmitted light intensity I is then obtained by integrationover θ_(0→φ):

$\begin{matrix}{I = {\int_{0}^{\varphi}{I_{0,\theta}R^{({\frac{2}{\tan \; \theta} - 1})}10^{- \frac{2\alpha}{\sin \; \theta}}}}} & (18)\end{matrix}$

For φ=π/₄ rad (45°), equation 18 can be numerically integrated as afunction of α, with a resolution of 0.56 mrad (0.1°) for various valuesof R and the results are shown as dashed line traces in FIG. 3. Thesecalculations assume that the initial intensity is the same at allangles, though this of course is not the case for a real LED. If it isassumed that the angular light intensity distribution of the LED isGaussian and has a full width half maximum of 0.26 rad (15°), b can bemuch higher because the total light throughput shifts to a higheraverage traversed pathlength. The overall limiting b can be less thanwhat is expected in a single-path multipass paradigm as embodied inequation 3.

In one aspect, the measurement cell 10 can comprise the light source 20,the detector 22 and a liquid core waveguide 36 configured to guide lightfrom the light source to the detector as illustrated in FIGS. 7 and 8.In this aspect, the light source can be coupled to an entrance end ofthe liquid core waveguide, and the detector can be coupled to an exitend of the liquid core waveguide. In use, light emitted from the lightsource 20 can travel through the liquid core waveguide. In anotheraspect, the liquid core waveguide can be a tube where light undergoestotal internal reflection at the inner or outer surface of the tube. Forexample, a water filled tube wherein the tube material has a refractiveindex less than that of water can behave as a liquid core waveguide. Ina further aspect, an entrance end wall 38 and an exit end wall 40 of theliquid core waveguide can be at least partially mirrored so that thelight can be reflected in the liquid core waveguide 36 as describedabove. The detector 22 can be positioned adjacent the exit end wall todetect light transmitted through the exit end wall.

With reference to FIG. 7, in one aspect, the entrance end wall 38 can beformed from an optical fiber bundle 42 comprising a plurality of fibersmirrored and for at least one central fiber 44 unmirrored. That is, thecentral fiber of the optical fiber bundle can be unmirrored to act asthe input aperture 24. Referring now to FIG. 8, in another aspect, theentrance end wall 38 can be formed from an optical fiber 46 having apartially mirrored face 48. In this aspect, the input aperture can be anunmirrored portion of the entrance end wall. For example, a centralportion of the face of the optical fiber can be unmirrored, and theremaining, surrounding portion of the face 48 of the optical fiber 46can be mirrored. The exit end wall 40 can be formed from an opticalfiber 50 having a mirrored exit face 52. For example and withoutlimitation, the exit end wall can be formed from a mirrored single coreoptical fiber.

Unlike other cells known and described in the literature, e.g., thePfund cell, White cell, the Herriott cell where light may undergomultiple reflections but the focused beam (where focus is maintained bythe use of concave mirrors) traces a single path, in this measurementcell 10, the deliberately divergent beam and flat surface nonfocusingminor can result in a highly nonlinear absorbance-concentrationbehavior. This nonlinearity effectively increases the dynamic range ofconcentration that can be measured.

EXPERIMENTAL SECTION

An experiment was conducted with two types of detection arrangements anddisposable polystyrene cuvettes in three designs. The detectors 22ranged from a high end commercial double-beam double monochromatorspectrometer capable of reading ±6.000 absorbance(http://www.shimadzu.com/an/spectro/uv/uv3600.html), to an inexpensivefiber optic 12-bit CCD spectrometer(http://oceanoptics.com/products/usb2000.asp, now obsolete). Thespectrophotometer measurements were made at the absorption maximum, witha slit width of 2 nm, except as stated. An identically prepared cell 10filled with water was used as the reference. With the CCD spectrometer,LEDs (center wavelengths of 617 and 522 nm) were used as the lightsource 20, generally just glued on the cell 10; the drive current was 50mA. The detector integration time was typically 1.5 seconds. The sameexperiment was repeated using a 633 nm laser as the source.

The following were the three cell designs: (1) the cell 10 was wrappedwith a single layer of reflective aluminized Mylar™ film (0.125 mmthick, sold as disposable emergency thermal blanket, www.sears.com). Theaperture 24 was made on the light entrance side 16 of the cellpositioned at the incident beam spot center (for the spectrophotometer,a rectangular 1×1.5 mm (w×h) aperture, for the fiber-optic CCD a 0.8 mmdiameter cutout). (2) Microscope slides were mirrored using a commercialsilvering solution (www.peacocklabs.com), these were affixed on thelight entrance and exit sides of the cell. By etching with HNO₃, a ˜0.5mm aperture 24 was made on the metaled side at the incident beam center.The glass sides of each mirror faced the cuvettes while the metaledsides were protected with a thin layer of transparent optical gradeepoxy. (3) The cell 10 was essentially identical to that in (2) exceptthat the optical fiber of the detector 22 was at a horizontal plane 1.8cm higher relative to the source aperture 24 (vertically offset, cf.FIG. 3B; this arrangement is hereinafter referred to as off-axis). Alsoin this off-axis case, light was read through a 0.5 mm bore 32 in theexit-side minor, unlike setup (2), where light was read through themirror itself, without a bore.

To make performance comparisons with and without reflective elements,the same type of cells, without any reflective elements, was used. Tocalculate effective path lengths ( b), the observed absorbance in thereflective cells was divided by the observed absorbance in the standardtransparent cells. However, because the absorbance in the standardtransparent cells was not directly measurable at very lowconcentrations, it was obtained by making Beer's law plots for each testdye (Bromocresol Green, Bromthymol Blue, and Erythrosine B (FD&C Red No.3) in the 0.010-1.000 absorbance range, respectively at 617, 615, and522 nm. The first two were prepared in 1 mM NaOH; this was also used asthe reference solution. Erythrosine was prepared in water. Standardlaboratory reagents were used. Diluted standards were typically preparedimmediately before use, at the lowest concentrations, often in themeasurement cell itself to prevent adsorption losses.

The Y-intercepts in the above Beer's law plots were statisticallyindistinguishable from zero, the linear relationship was excellent(r²≧0.999), and the molar absorptivities (∈) calculated from the slopewere in excellent agreement with literature values. The absorbance inthe conventional cell expected for the dye at the low concentrations wasthen computed from ∈.

Reflectivities were measured typically at three wavelengths, using 633nm (He—Ne), 532 nm (frequency doubled Nd-YAG) and 405 nm (InGaN) lasersand a laser power meter (www.sperdirect.com).

Cell types 1 and 2 were examined with both the spectrophotometer and theCCD spectrometer, cell type 3 was examined only with the latter.

Results

Concentration—Apparent Absorbance Relationships.

FIG. 5 shows Beer's law behavior of bromocresol green at pH 12 inconventional linear as well as logarithmic axes. The beginning andending values of both axes were the same such that the starting and theending points in both the linear and logarithmic plots are coincident.Two representative cases are shown: (a) in a silver-mirrored cell (type2) with a 633 nm laser and a CCD array detector; and (b) a reflectivemylar-wrapped cell with the UV 3600 spectrophotometer as the source (617nm, 2 nm slit width) and detector. In FIG. 5A, referring to the linearaxes there are three distinct regions: (i) a very steep A vs C linearrelationship at the low concentration end, (ii) a less steep linearportion at the higher concentration end, and (iii) a transition regionof low slope in-between. The ratio of the two slopes in (i) and (ii) isthe plateau value of the pathlength amplification factor in FIG. 2 andthe plateau is attained in region (i). The same three regions are alsoobserved in the log-log plot; the noteworthy item is that log A-log Cslope is very close to unity in both regions (i) and (ii). We refer tothis as type A behavior, in which plateau value is reached.

FIG. 5B shows the other type of behavior: while this also exhibits threeregions and the log A-log C relationship has a unit slope at the higherconcentration end, the log A-log C slope is less than unity at the lowerconcentration end. In this type of behavior (hereinafter called type B)a plateau of pathlength amplification factor is not observed; this willbecome more apparent in the next section. The data in FIG. 5B also showa greater standard deviation relative to that in FIG. 5A because of poorlight throughput resulting from a broadband source and a narrow slitwidth. An increased slit width increases light throughput and reducesthe standard deviation of the measurements (it also increases theeffective pathlength amplification factor); all LOD data for thespectrophotometric measurements are therefore reported for measurementsmade with a 20 nm slit width.

The Limiting Value of b.

Two basic types of concentration vs. b behavior were observed. In thefirst type of behavior (case A—this includes all experiments with theCCD spectrometer and laser or narrow angle LED sources), the resultswere in accordance with the theoretical expectations depicted in FIGS.2, 3, and 4 b—namely, a limiting plateau value of b was attained orapproached (FIG. 5 a). The observed

$\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}$

value was also quantitatively close to the theoretically expected bvalues. For example, in the laser source experiment (green trace, FIG. 5a) the

$\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}$

was ˜55, which would be expected for mirrors with R=0.982, the measuredR was 0.99±0.01. The identical experiment, except with a more divergentLED source, can be expected to have a lower limiting b because ofearlier series truncation,

$\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}$

in this case (red trace, FIG. 5 a) was ˜40. For a 15° FWHM LED, forR=0.99 case A, a

$\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}$

of 30 was calculated (FIG. 2) and the experimental

$\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}$

value was ˜35. For R=0.93,

$\lim\limits_{\alpha\rightarrow 0}\overset{\_}{b}$

was expected to be 13.8 (equation 4) and the observed value at thelowest measured concentration was 11.3±3.1.

In case B (the second type of behavior), there was no indication that alimiting plateau value of b was being approached. Rather, b increasedexponentially with decreasing log (concentration), within the lowerlimits of the analyte concentrations that could be reliably measured, asshown in FIG. 5 b. All spectrophotometer experiments exhibited this typeof behavior. Case B behavior can be associated with beam divergence andcan also be observed in the CCD spectrometer experiments if wide angleemitters are used, instead of a narrow beam (15° FWHM) LED. Increasingspectrophotometer slit width increases beam divergence and as FIG. 5 bshows, the observed effect is more pronounced. Although a quantitativerelationship has not been formulated herein, this behavior can beexpected if the number of terms that matter in a truncated seriesincreases with decreasing α. For similar reasons (because extinction ismore rapid) this effect is also expected with decreasing mirrorreflectivity. The observed absorbance (directly proportional tob)—concentration relationship can be different in case A than in case B.In case A, if b has reached a constant limiting value, then absorbanceis directly proportional to concentration and this value of b issynonymous with the term “Cavity Enhancement Factor” and Lambert-Beer'slaw can be followed in the b plateau region. On the other hand, when bincreases exponentially with decreasing log C, Lambert-Beer's law can benot followed. log A (or log b) is linearly related to log C in thisregime, as shown in the inset of FIG. 5 b. Of course, in case A, when Ais linearly related to C with a near-zero intercept, log A can alsonecessarily be linearly related to log C. In contrast, in case B, A isnot necessarily linearly related to C.

Increasing the absorbance has no practical value unless there is actualgain in the limit of detection (LOD). Tables I and II list the observedexperimental data for the CCD spectrometer and the spectrophotometer,respectively that demonstrate that real improvements in LODs areachieved.

The LODs were calculated by two different methods. One method assumed alinear relationship between A and C and the other method assumed alinear relationship between log A and log C. The absorbance of the blankwas measured five times and the LOD was taken to be the C valuecorresponding to three times the standard deviation of the blank on theA vs C or log A vs log C plot, respectively. The two calculated LODvalues differed significantly only in case B, when A was not linearlyrelated to C. These cases are indicated in the tables with asterisks andin these cases the logarithmic relationship based LOD can be moreappropriate. For comparison, measurements were also made in aconventional (no mirrors) cell. The linear relationship between A and Cwas confirmed in the absence of mirrors and the LOD was computed fromslope of the terminal two points above the LOD and three times theuncertainty of the blank as indicated above. In all cases the LODimproved in the cavity enhanced measurements, in some cases by more thanan order of magnitude.

TABLE I Summary of Results, CCD Spectrometer Detector Lowest Effectivetested pathlength at concn., lowest concn., LOD₁, LOD₂, Cell typeAnalyte Reflectivity nM^(a) b cm nM^(b) nM^(c) Conventional, Bromocresolgreen, NA 1 36 no mirror ε = 39100 M⁻¹ cm⁻¹ Type 1: ReflectiveBromocresol green 0.93 @ 633 nm 10  11 ± 0.5 8 8 Mylar, entranceaperture, 617 nm LED Type 2: Glass Bromocresol green 0.99 @ 633 nm 10 41 ± 3.2 4 4 mirror, entrance aperture, 617 nm LED Type 2: 633 nmBromocresol green 0.99 @ 633 nm 10 55 ± 16 3 3 laser source Type 3,Off-axis Bromocresol green 0.99 @ 633 nm 10  33 ± 4.7 12 9 617 nm LEDConventional, Erythrosine NA 25 1 46 no mirror ε = 87,500 Type 1:Reflective Erythrosine 0.89 @ 532 nm 25 8.2 ± 0.8 5 4 Mylar, entranceaperture, 522 nm LED* Type 2: Glass Erythrosine 0.91 @ 532 nm 25 8.8 ±0.3 4 0.4 mirror, entrance aperture, 522 nm LED* ^(a)lowestconcentration measured; not considered if below the LOD ^(b)Based on theterminal slope of the Absorbance vs concentration curve ^(c)Based on theterminal slope of the log (Absorbance) vs log (concentration) curve *atlow concentrations, log A varies linearly with log C.

TABLE II Summary of Results, UV Spectrophotometer Detector* LowestEffective tested pathlength Concn, at lowest LOD₁, LOD₂, Cell typeAnalyte Reflectivity nM^(a) concn, b nM^(b) nM^(c) Conventional,Bromocresol green @ 47 no mirror 617 nm Type 1: Reflective BromocresolGreen @ 0.93 @ 633 nm 10 16 ± 4.6 33 5 Mylar, Entrance 617 nm ApertureType 2: Glass Bromocresol green @ 0.99 @ 633 nm 25 95 ± 23  6 0.4Mirror, Entrance 617 nm Aperture Conventional, Erythrosine @ 13 nomirror 522 nm Type 1: Reflective Erythrosine @ 0.89 @ 532 nm 12.5 26 ±4.3 8 7.3 Mylar, Entrance 522 nm Aperture Type 2: Glass Erythrosine @0.91 @ 532 nm 12.5 48 ± 12  10 7.2 Mirror, Entrance 522 nm Aperture *atlow concentrations, log A varies linearly with log C. ^(a)lowestconcentration measured; not considered if below the LOD ^(b)Based on theterminal slope of the Absorbance vs concentration curve ^(c)Based on theterminal slope of the log (Absorbance) vs log (concentration) curve

In summary, in one aspect, a generally applicable (regardless of theexact values of R, T or α), mathematical foundation for cavity enhancedabsorption measurements has been provided. In another aspect, an inputaperture 24 for the cell 10 has been provided which can be of value in asituation where beam divergence is present. This can result in greaterlight throughput without significant loss of cavity enhancement and thusreduces the need for increased source brightness for sufficient light toreach the detector for shot noise limited operation. This can provide anattractive means to make sensitive absorbance measurements with goodlight throughput in systems where the physical pathlength is limited,e.g., in capillary systems. It has also been shown that similarnonlinear absorbance amplification can result from the multipath effectin a reflecting cavity that contains an entrance and additionally anoptional exit bore that do not necessarily directly face each other.

Although several aspects of the invention have been disclosed in theforegoing specification, it is understood by those skilled in the artthat many modifications and other aspects of the invention will come tomind to which the invention pertains, having the benefit of the teachingpresented in the foregoing description and associated drawings. It isthus understood that the invention is not limited to the specificaspects disclosed hereinabove, and that many modifications and otheraspects are intended to be included within the scope of the appendedclaims. Moreover, although specific terms are employed herein, as wellas in the claims that follow, they are used only in a generic anddescriptive sense, and not for the purposes of limiting the describedinvention.

What is claimed is:
 1. A measurement cell for absorbance measurements ofa medium positioned in a cavity of the cell comprising; an entranceminor forming at least a portion of a first wall of the cell, wherein aninput aperture is defined in a portion of the entrance minor; an exitmirror forming at least a portion of a second wall of the cell that isopposed to the first wall; a light source configured to input a beam oflight into the cavity of the cell through the input aperture; and adetector positioned outside of the cell adjacent to the exit mirrorconfigured to convert light detected exiting the cell through the exitminor into an electrical signal for further analysis by a processor. 2.The cell of claim 1, wherein the beam of light is a divergent beam. 3.The cell of claim 2, wherein at least a portion of the light enteringthe cell through the input aperture travel across the cavity andreflects off the exit minor and back towards the entrance mirror.
 4. Thecell of claim 3, wherein a portion of the light reflected back towardsthe entrance mirror is lost from the cell through the input aperture. 5.The cell of claim 4, wherein the input aperture is sized so that thelost light is small relative to a total area of the beam of light. 6.The cell of claim 1, wherein the input aperture is substantiallycircular in cross-sectional shape.
 7. The cell of claim 6, wherein theinput aperture has a substantially constant diameter.
 8. The cell ofclaim 6, wherein a diameter of the input aperture decreases as theaperture extends from an outer surface of the entrance mirror to aninner surface of the entrance mirror.
 9. The cell of claim 1, whereinthe medium is in a gaseous phase.
 10. The cell of claim 1, wherein thedetector is positioned outside of the cell adjacent to the exit minor ata different horizontal plane relative to the input aperture.
 11. Thecell of claim 10, wherein a longitudinal axis of the detector is offsetfrom a longitudinal axis of the input aperture a predetermined amount.12. The cell of claim 1, wherein an exit bore is defined in a portion ofthe exit minor, and wherein the detector is positioned adjacent the exitbore.
 13. The cell of claim 1, wherein the entrance minor is anonfocusing mirror.
 14. The cell of claim 13, wherein the exit mirror isa nonfocusing mirror.
 15. The cell of claim 1, wherein the entranceminor is substantially planar.
 16. The cell of claim 15, wherein theexit mirror is substantially planar.
 17. The cell of claim 1, whereinthe cell further comprises a liquid core waveguide, wherein the lightsource is coupled to an entrance end of the liquid core waveguide, andthe detector is coupled to an exit end of the liquid core waveguide. 18.The cell of claim 17, wherein the entrance minor comprises an opticalfiber having a mirrored entrance face, and wherein the exit mirrorcomprises an optical fiber having a mirrored exit face.
 19. The cell ofclaim 18, wherein the input aperture is defined in the mirrored entranceface.
 20. A method of improving light throughput in absorptionspectrometry comprising: providing a measurement cell for absorbancemeasurements of a medium positioned in a cavity of the cell comprising;an entrance minor forming at least a portion of a first wall of thecell; an exit mirror forming at least a portion of a second wall of thecell that is opposed to the first wall; a light source; and a detectorpositioned outside of the cell adjacent to the exit mirror configured toconvert light detected exiting the cell through the exit minor into anelectrical signal for further analysis by a processor; defining an inputaperture in a portion of the entrance mirror; and inputting a divergentbeam of light into the cavity of the cell through the input aperture.21. A method of increasing the detection sensitivity of absorptionspectroscopy comprising: providing a measurement cell for absorbancemeasurements of a medium positioned in a cavity of the cell comprising;an entrance minor forming at least a portion of a first wall of thecell; an exit mirror forming at least a portion of a second wall of thecell that is opposed to the first wall; a light source; and a detectorpositioned outside of the cell adjacent to the exit mirror configured toconvert light detected exiting the cell through the exit minor into anelectrical signal for further analysis by a processor; defining an inputaperture in a portion of the entrance mirror; and inputting a divergentbeam of light from the light source into the cavity of the cell throughthe input aperture.